Das Clay-Institut hatte im Jahr 2000 sieben Preise zu je 106 $ für die Lösung offener mathematischer Probleme ausgelobt.
Eines der 7 Probleme war die Poincaré-Vermutung. (Hier ein Link zur Ausschreibung und den anderen Problemen.)
Bekanntlich wurde die Poincaré-Vermutung 2003 in einer Serie von ArXiv-Preprints von Perelman bewiesen, ausführlichere Darstellungen des Beweises u.a. von Kleiner-Lott und Morgan-Tian erschienen in den letzten Jahren, inzwischen gilt der Beweis als allgemein akzeptiert.
Wie heute bekanntgegeben wurde, wird das Clay-Institut den Millenium-Preis für die Lösung der Poincaré-Vermutung an Perelman verleihen. (Auf Basis einer Empfehlung des Special Advisory Committee, bestehend aus Donaldson, Gabai, Gromov, Tao und Wiles.)
Aus der Mitteilung des Clay-Instituts:
The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads:
[…]
The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.
[…]
Formulated in 1904 by the French mathematician Henri Poincaré, the conjecture is fundamental to achieving an understanding of three-dimensional shapes (compact manifolds). The simplest of these shapes is the three-dimensional sphere. It is contained in four-dimensional space, and is defined as the set of points at a fixed distance from a given point, just as the two-dimensional sphere (skin of an orange or surface of the earth) is defined as the set of points in three-dimensional space at a fixed distance from a given point (the center).
Since we cannot directly visualize objects in n-dimensional space, Poincaré asked whether there is a test for recognizing when a shape is the three-sphere by performing measurements and other operations inside the shape. The goal was to recognize all three-spheres even though they may be highly distorted. Poincaré found the right test (simple connectivity, see below). However, no one before Perelman was able to show that the test guaranteed that the given shape was in fact a three-sphere.
In the last century, there were many attempts to prove, and also to disprove, the Poincaré conjecture using the methods of topology. Around 1982, however, a new line of attack was opened. This was the Ricci flow method pioneered and developed by Richard Hamilton. It was based on a differential equation related to the one introduced by Joseph Fourier 160 years earlier to study the conduction of heat. With the Ricci flow equation, Hamilton obtained a series of spectacular results in geometry. However, progress in applying it to the conjecture eventually came to a standstill, largely because formation of singularities, akin to formation of black holes in the evolution of the cosmos, defied mathematical understanding.
Perelman’s breakthrough proof of the Poincaré conjecture was made possible by a number of new elements. Perelman achieved a complete understanding of singularity formation in Ricci flow as well as the way parts of the shape collapse onto lower-dimensional spaces. He introduced a new quantity, the entropy, which decreases as time increases during Ricci flow, signaling an increase in the degree of geometric order in the underlying shape. He introduced a related local quantity, the L-functional, and he used the theories of Cheeger and Aleksandrov to understand limits of spaces changing under Ricci flow. He was also able to show that the time between formation of singularities could not become smaller and smaller, with singularities becoming spaced so closely – infinitesimally close – that the Ricci flow method would no longer apply. Perelman deployed his new ideas and methods with great technical mastery and described the results he obtained with elegant brevity. Mathematics has been deeply enriched.
Etwas ausführlicher, auch zur ‘Entwicklungsgeschichte’, ist die Darstellung auf https://www.claymath.org/poincare/continuation.html.
Und noch ein Link: ein Video eines kurzen populärwissenschaftlichen Vortrages von McMullen zum Beweis der Poincaré-Vermutung.
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